Public-school teachers are suing the city to keep control of their lesson plans away from supervisors — a move that scored an “F” from reformers.

The United Federation of Teachers filed a lawsuit in Manhattan Supreme Court Thursday asking a judge to confirm a little-noticed May arbitrator’s decision declaring that teachers — not principals — are in charge of deciding what goes into lesson plans.

But reformers fear that if the union wins the suit, it could have a devastating effect on students.

“It’s outrageous that teachers believe they don’t need to share their lesson plans with the principal beforehand,” said Mona Davids, president of the New York City Parents Union, who plans to oppose the suit. “How is a principal supposed to ensure students are receiving a high-quality education?”

The union is taking the extra legal step to strengthen its hand on the issue. Even if teachers have control now, a judge’s decision would be stronger than an arbitrator’s under a challenge.

“This lawsuit is union boss Mike Mulgrew’s latest attempt to gum up the works at the expense of our kids,” thundered Jenny Selis, head of reform-minded nonprofit StudentsFirstNY. “The only reason for this frivolous ­legal action is to protect the poorly performing teachers who don’t properly plan and aren’t serving our kids well.”

The UFT didn’t even want principals to be able to collect lesson plans, but arbitrator Deborah Gaines, in a May 16 ruling, denied the union’s request.

During the arbitration hearing, a Department of Education representative noted that the union defended a teacher who “merely strung together a list of song titles” and called that a lesson plan.

Total lack of oversight “would, in fact, diminish the professionalism of teachers,” the DOE argued.

A spokeswoman for the principals union said the organization isn’t planning to challenge the move, since principals can still see the plans and make suggestions.

Mulgrew defended the union’s case to The Post. He said in a statement that teacher control over lesson plans “helps to reduce the amount of paperwork ­required of teachers.”

He added that teachers themselves should be empowered to decide what happens in the classroom.

A spokeswoman for the city’s Law Department said, “We will review the petition upon receipt.”